A Fourier transform for the quantum Toda lattice
Abstract We answer a question of V. Drinfeld by constructing an ‘algebraic Fourier transform’ for the quantum Toda lattice of a complex reductive algebraic group G, which extends the classical ‘algebraic Fourier transform’ for its subalgebra...
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| Outros autores: | |
| Formato: | Artigo |
| Idioma: | English |
| Publicado: |
Springer International Publishing
2021
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| Acceso en liña: | https://hdl.handle.net/1721.1/131565 |
| _version_ | 1826198482335039488 |
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| author | Lonergan, Gus |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Lonergan, Gus |
| author_sort | Lonergan, Gus |
| collection | MIT |
| description | Abstract
We answer a question of V. Drinfeld by constructing an ‘algebraic Fourier transform’ for the quantum Toda lattice of a complex reductive algebraic group G, which extends the classical ‘algebraic Fourier transform’ for its subalgebra
$$D(T)^W$$
D
(
T
)
W
of Weyl group invariant differential operators on a maximal torus. The proof is contained in Sect. 2 and relies on a result of Bezrukavnikov–Finkelberg realizing the quantum Toda lattice as the equivariant homology of the dual affine Grassmannian; the Fourier transform boils down to nothing more than the duality between homology and cohomology. In Sect. 3, we compare our result with a related result of V. Ginzburg, and explain the apparent discrepancy by showing that W-equivariant quasicoherent sheaves on
$${{\mathrm{\mathfrak {t}}}}^*$$
t
∗
descend to
$${{\mathrm{\mathfrak {t}}}}^*//W$$
t
∗
/
/
W
if they descend to
$${{\mathrm{\mathfrak {t}}}}^*/\langle s_i\rangle $$
t
∗
/
⟨
s
i
⟩
for every simple reflection
$$s_i$$
s
i
of W. |
| first_indexed | 2024-09-23T11:05:38Z |
| format | Article |
| id | mit-1721.1/131565 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T11:05:38Z |
| publishDate | 2021 |
| publisher | Springer International Publishing |
| record_format | dspace |
| spelling | mit-1721.1/1315652023-09-27T20:04:27Z A Fourier transform for the quantum Toda lattice Lonergan, Gus Massachusetts Institute of Technology. Department of Mathematics Abstract We answer a question of V. Drinfeld by constructing an ‘algebraic Fourier transform’ for the quantum Toda lattice of a complex reductive algebraic group G, which extends the classical ‘algebraic Fourier transform’ for its subalgebra $$D(T)^W$$ D ( T ) W of Weyl group invariant differential operators on a maximal torus. The proof is contained in Sect. 2 and relies on a result of Bezrukavnikov–Finkelberg realizing the quantum Toda lattice as the equivariant homology of the dual affine Grassmannian; the Fourier transform boils down to nothing more than the duality between homology and cohomology. In Sect. 3, we compare our result with a related result of V. Ginzburg, and explain the apparent discrepancy by showing that W-equivariant quasicoherent sheaves on $${{\mathrm{\mathfrak {t}}}}^*$$ t ∗ descend to $${{\mathrm{\mathfrak {t}}}}^*//W$$ t ∗ / / W if they descend to $${{\mathrm{\mathfrak {t}}}}^*/\langle s_i\rangle $$ t ∗ / ⟨ s i ⟩ for every simple reflection $$s_i$$ s i of W. 2021-09-20T17:20:25Z 2021-09-20T17:20:25Z 2018-06-05 2020-09-24T21:10:52Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/131565 en https://doi.org/10.1007/s00029-018-0419-x Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer International Publishing AG, part of Springer Nature application/pdf Springer International Publishing Springer International Publishing |
| spellingShingle | Lonergan, Gus A Fourier transform for the quantum Toda lattice |
| title | A Fourier transform for the quantum Toda lattice |
| title_full | A Fourier transform for the quantum Toda lattice |
| title_fullStr | A Fourier transform for the quantum Toda lattice |
| title_full_unstemmed | A Fourier transform for the quantum Toda lattice |
| title_short | A Fourier transform for the quantum Toda lattice |
| title_sort | fourier transform for the quantum toda lattice |
| url | https://hdl.handle.net/1721.1/131565 |
| work_keys_str_mv | AT lonergangus afouriertransformforthequantumtodalattice AT lonergangus fouriertransformforthequantumtodalattice |